We address some issues concerning the relationship between metaphor and creativity, arising from an AI project (ATT-Meta) that has developed a partially-implemented theory of metaphor understanding. The central claim is that, while metaphor is plausibly based partly on structured mappings (analogies) between a target subject matter (the topic actually being addressed) and some source subject matter, creative metaphorical phraseology very often does not require construction of new mapping links to handle source aspects used by the phraseology but not handled by existing mappings. Thus, such phraseology often rests on substantial NON-parallelism (non-analogy) between source and target. Rather, the unmapped source aspects serve only to indirectly control what information is transferred between source and target by already-known mapping links; and the phraseology, while itself creative, is not causing a grandly creative new look at the target. However, the approach explains relatively easily the nature of much creative metaphor and how it can be understood, and has some implications for creativity more generally.